arXiv subjects

Qidong Guo

Publications and source records attributed to Qidong Guo.

3 recordsLinked to original sources

arXiv2026

Circle-like concentrated solutions for two-component Bose-Einstein condensates

We investigate the normalized solutions of the following two-component Bose-Einstein condensates (BEC) system \begin{equation}\left\{ \begin{split} -Δu + (λ+P(x))u &= αu^3 +βuv^2, && \text{in } \mathbb{R}^2,\\-Δv + (λ+Q(x))v &= γv^3 +βu^2 v, && \text{in } \mathbb{R}^2, \end{split} \right.\end{equation} with $L^2$-constraint $$\int_{\mathbb{R}^2}(u^2+v^2)\,dx = 1.$$ For any $α>0$, $γ> 0$ and $\ β\in (-\sqrt{αγ},0)\cup(0,\min \{α,γ\})\cup \left(\max \{α,γ\} , + \infty\right)$, we establish the existence of synchronized solutions concentrating on high-dimensional subsets of $\mathbb{R}^2$ by employing a finite-dimensional reduction method combined with some local Pohozaev identities. More precisely, we construct vector radial solutions that concentrate on circles when $ \frac{α+ γ- 2β}{αγ- β^2}$ tends to zero. Our results fill the blank in the system for high-dimensional concentrated normalized solutions.

math.AP↗

arXiv2025

Normalized Solutions for nonlinear Schrödinger-Poisson equations involving nearly mass-critical exponents

We study the Schrödinger-Poisson-Slater equation \begin{equation*}\left\{\begin{array}{lll} -Δu + λu + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, \, \text{ in } \mathbb{R}^{3},\\[2mm] \int_{\mathbb{R}^3}u^2 \,dx= a,\,\, u > 0,\,\, u \in H^{1}(\mathbb{R}^{3}), \end{array} \right. \end{equation*} where $λ$ is a Lagrange multiplier, $V(x)$ is a real-valued potential, $a\in \mathbb{R}_{+}$ is a constant, $ p_{\varepsilon} = \frac{10}{3} \pm \varepsilon$ and $\varepsilon>0$ is a small parameter. In this paper, we prove that it is the positive critical value of the potential $V$ that affects the existence of single-peak solutions for this problem. Furthermore, we prove the local uniqueness of the solutions we construct.

math.AP↗

arXiv2024

Construction of solutions to a nonlinear critical elliptic system via local Pohozaev identities

In this paper, we investigate the following elliptic system with Sobolev critical growth $-Δu+P(|y'|,y'')u=u^{2^*-1}+\fracβ{2} u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}},\ y\in R^N$, $-Δv+Q(|y'|,y'')v=v^{2^*-1}+\fracβ{2} v^{\frac{2^*}{2}-1}u^{\frac{2^*}{2}}$, $y\in R^N ,u,v>0,u,\,v\in H^1(R^N), $ where~$(y',y'')\in R^2 \times R^{N-2}$, $P(|y'|,y''), Q(|y'|,y'')$ are bounded non-negative function in $R^+\times R^{N-2}$, $2^*=\frac{2N}{N-2}$. By combining a finite reduction argument and local Pohozaev type of identities, assuming that $N\geq 5$ and $r^2(P(r,y'')+κ^2Q(r,y''))$ have a common topologically nontrivial critical point, we construct an unbounded sequence of non-radial positive vector solutions of synchronized type, whose energy can be made arbitrarily large. Our result extends the result of a single critical problem by [Peng, Wang and Yan,J. Funct. Anal. 274: 2606-2633, 2018]. The novelties mainly include the following two aspects. On one hand, when $N\geq5$, the coupling exponent $\frac{2}{N-2}<1$, which creates a great trouble for us to apply the perturbation argument directly. This constitutes the main difficulty different between the coupling system and a single equation. On the other hand, the weaker symmetry conditions of $P(y)$ and $Q(y)$ make us not estimate directly the corresponding derivatives of the reduced functional in locating the concentration points of the solutions, we employ some local Pohozaev identities to locate them.

math.AP↗